Magnetic fields in discs: what can be learned from infrared and mm polarimetry?

Abstract

Polarimetric imaging in the infrared and submm offers the possibility of identifying magnetic field configurations in astronomical objects. To test this conjecture a set of field geometries within internally heated discs has been modelled and polarization transfer followed for a range of view angles with respect to the disc axis. The wavelength range considered is from the mid-infrared to submm, the dominant polarization processes then being only dichroic emission and absorption by aligned grains. A sample of the resulting polarization images is presented and their salient features discussed. There are obvious, and some not so obvious, associations of polarization structure with the parent model field and, while these are not always unique, they will usually lead to strong constraints on the field configuration. For star formation regions the polarization structure is likely to be on a small spatial scale and then the full potential of this technique must await the advent of millimetre synthesis systems.

magnetic fields, techniques: polarimetric, stars: formation, dust, extinction, infrared: general

Introduction

Magnetic fields almost certainly play an important role in star formation. So far the bulk of the effort in attacking this problem has been theoretical through the application of MHD to rotating accretion discs (e.g. Blandford & Payne 1982; Uchida & Shibata 1985; Pudritz & Norman 1986; McKee et al. 1993). These studies have yielded a range of possible field distributions which variously assist or mediate both outflows and accretion. Currently the most promising way to observe the orientation of magnetic fields in the dense cores of molecular clouds where star formation is taking place is through the study of polarization produced by aligned dust grains in these places. In the mid-infrared, emission from warm grains can indicate the fields close to the central source(s) and absorption by cooler grains probes the fields along the line of sight, while, in the far-infrared and submm, polarized emission will reveal fields throughout the extent of the clouds.

Observational work has so far been sparse and, apart from a few exceptions, mainly by low-resolution (≳15 arcsec) polarimetric imaging in the far-infrared and submm and medium resolution (≃1 arcsec) polarimetric imaging and spectropolarimetry in the mid-infrared. However, current trends in detector technology, large telescope accessibility and synthesis systems will soon enable the production of sub-arcsec resolution polarimetric and spectropolarimetric images of star formation regions from the near infrared through the submm. Such studies should produce a breakthrough in this subject.

The interpretation of these observational data, depending in many cases on both polarized emission and absorption along the lines of sight, is not trivial and in anticipation of this the present modelling work has been undertaken with two main aims in mind: (i) to establish the ability of polarimetric imaging (at a range of wavelengths) to discriminate between various field geometries and (ii) to aid in the interpretation of observations. To avoid complications introduced by scattering we have concentrated only on polarization arising from the dichroic emission and absorption from aligned grains, and this effectively restricts the study to the mid through far-infrared and submm.

Previous modelling of the polarization resulting from dusty magnetized structures has usually been limited to optically thin dichroic emission (e.g. Wardle & Königl 1990) although Akeson & Carlstrom (1997) have modelled the dichroic emission from an hourglass field configuration in which the opacity is not insignificant. Desch & Roberge (1997) have repeated the modelling of Wardle & Königl (1990), but using ambipolar diffusion as the grain alignment mechanism. More recently the effect of the transfer of polarization from a centrally heated dusty disc (Efstathiou, McCall & Hough 1997) has been investigated. In this study the aligning magnetic field was parallel to the disc axis and the resulting polarization from 5 to 100 μm found for a range of view angles to the disc plane. Here we adopt a similar procedure but also look at the spatial distribution of polarization and its spectrum over a wavelength range from the mid-infrared to the submm, and introduce a variety of field configurations.

Outline of method

The approach that has been taken is to adopt a specific disc geometry with a temperature distribution determined by radiation transfer from a central luminous source. The dust grains are taken as uniformly aligned with average spin axes parallel to the local magnetic field and the transfer of polarization along various lines of sight derived; the effect of grain alignment on radiative transfer is neglected.

Here the discs are optically thick in the visible and are uniformly flared. The temperature distributions were calculated with the codes of Efstathiou & Rowan-Robinson (1990, 1994, 1995), which incorporate a range of sizes of silicate and graphite grains that have the extinction and scattering properties described in Efstathiou, Rowan-Robinson & Siebenmorgen (2000). This grain model is based on that of Siebenmorgen & Krugel (1992), which includes a mixture of astronomical silicate grains (with the dielectric constants of Draine & Lee 1984), amorphous carbon grains and small graphites and polycyclic aromatic hydrocarbons (PAHs). In the present study only the grains of radius 0.06 μm are aligned. Here radius refers to that of a sphere of the same volume as the oblate grains; grains of this size will be capable of alignment and will not mute the spectral features. The radiative transfer models of Efstathiou & Rowan-Robinson (1990, 1995) are dimensionless, as is the usual practise in these kind of calculations (e.g. Rowan-Robinson 1980, Ivezic & Elitzur 1997) and does not mean any loss of generality, as the equation of radiative transfer itself can be cast in dimensionless form. The models assume that the disc is illuminated by a central blackbody source at temperature T_s and radius r_s, which creates a dust-free cavity of radius r₁ inside which the grains sublimate if they exceed a temperature T₁. In a multi-grain medium each grain species will have a different sublimation radius in which case r₁ refers to the minimum inner radius usually corresponding to the largest grains in the mixture. The parameters to which the temperature and spectral energy distributions are most sensitive are the optical depth to the centre of the cloud, the density distribution of the dust, usually assumed to be a power law , and the ratio of the inner and outer disc radii . Here we assume , and . The dependence of the temperature distribution on the assumed stellar temperature is actually very weak (Rowan-Robinson 1980) as long as is adjusted to keep T₁ constant.

These models are intended as an indication of how temperature distributions in dense discs (optically thick in the mid-infrared with , a few, along a line of sight within the flare angle to the central source) affect the resultant polarization and illustrate general trends rather than specific predictions, which will depend on the characteristics of individual sources. The models correspond roughly to Class 1 protostars in the usually assumed classification scheme (Lada & Wilking 1984) while the models can be thought of as two-dimensional extensions of the compact H ii region models of Efstathiou & Rowan-Robinson (1994), which were quite successful at fitting the available data. The temperature in these models decreases outwards so that they will be a poor representation of the earliest phases of protostellar collapse, Class 0 (Andre, Ward-Thompson & Barsony 1993), where the temperatures may increase outwards as the dominant heating is from the interstellar radiation field. Models of this type will be presented in future work.

Two of the temperature solutions have been used to develop polarization models: the flared discs in both have half-angle 30°, one has uniform density and the other has a r^−1.5 density dependence; the centre-to-edge visual extinction is usually 50 mag. As the Efstathiou models are dimensionless, the introduction of a physical scale requires the specification of source luminosity: if the central source is a 10⁴-L_⊙ protostar then the radial scale will be about per scale unit, or about 0.02 arcsec at a distance of 1 kpc. At such a distance the whole disc diameter subtends an angle of 40 arcsec, equivalent to about 50 000 au or 0.25 pc.

Although the grain temperatures are derived from the radiative transfer code of Efstathiou using the grain extinction properties defined by Efstathiou, Rowan-Robinson & Siebenmorgen (2000) to model their polarizing properties, they are taken as having the dielectric properties of ‘astronomical silicate’ (Lee & Draine 1985; Draine & Lee 1984) are used to define the polarizing properties of the grains. While the fit these dielectric functions give for the 10- and 20-μm features is imperfect (e.g. Aitken, Smith & Roche 1989; Greenberg & Li 1996), they reproduce the spectral profiles of polarization adequately for the present purpose.

A number of studies have shown that such grains do not need to be very aspheric to produce the observed values of polarization and in this work they are taken as oblate spheroids with axial ratio 0.8. The cross sections, and (e.g. Lee & Draine 1985), where C_⊥ and C_∥ are the grain cross-sections for radiation E vectors perpendicular and parallel to the grain symmetry axis respectively, are found for this geometry in the Rayleigh limit (e.g. Draine & Lee 1984). In taking the expression for C_abs we ignore the very small change of extinction resulting from alignment and to this approximation the derived surface brightness will be independent of the field configuration.

The grain number density for the polarization studies is normalized to give an approximate extinction of along a radius to the central source at the origin. For simplicity, and to reduce the number of parameters, in the majority of the polarization models only a single grain component consisting of the 0.06-μm silicate dust component in the Efstathiou model is used. The grains are assumed to be partially aligned with their spin axes along the ambient magnetic field with uniform degree of alignment given by . Here is the Rayleigh reduction factor (Greenberg 1968), where β is the precession angle between the grain spin axis and the magnetic field and 〈〉 denotes taking the average; for grain spins completely aligned along the field, for alignment normal to the field and for randomly oriented grains. The grain spin axes are taken to be those of the largest moment of inertia of the grain (e.g. Roberge 1996; Dolginov 1972); for oblate grains this is the symmetry axis. In the dense parts of quiescent molecular clouds there is evidence that grains are not aligned (Goodman et al. 1995), but in other situations grain alignment appears to be ubiquitous and fairly uniform (e.g. Hildebrand et al. 1999; Smith et al. 2000): the assumption of uniform alignment is not unreasonable as a first approximation for the environs of active star formation considered here. In that case and in the submm and beyond, where the polarization will be emissive and to a good approximation optically thin, a uniform magnetic field will give rise to a uniform polarization, in both position angle and amplitude irrespective of density changes. Thus the observation of uniform polarization will imply a uniform magnetic field.

When only a single dust component is considered there is no allowance for dilution by other possibly non-aligned grain species. This omission can affect the degree of polarization and its spectrum but the position angle should be little affected. In a subset of models that explore a more restricted range of fields and viewing angles we have added non-aligned grain components of different size and hence different temperature distributions.

A rectangular grid of parallel lines of sight equispaced about the disc centre as origin and at various inclination angles, i, to the disc mid-plane is set up; the temperature is interpolated linearly along these lines at incremental points spaced proportional to the distance from the origin. For a set of discrete wavelengths from the mid-infrared through to the submm the grid size is to provide detailed polarimetric images, and between 5 and 1000 μm a grid is used to provide low-resolution spectropolarimetry. The disc is described in cylindrical coordinates, r, z (symmetry axis) and φ, and the central line of sight is treated in more detail because of the strong radial dependence of temperature (and density in the non-uniform case), and here an average of more closely spaced lines of sight within a ‘beam’ radius of one third of the grid spacing is used. Similarly, because the r and φ components of the magnetic fields used often vanish on the equatorial plane, averages across this plane are taken for small view angles.

Most of the magnetic field configurations are taken as simple linear functions of z and r within the disc: , and . We are not concerned here with fields outside the disc, although we may imagine that all except the purely azimuthal and the dipole tend towards being axial after leaving the disc. The field directions are transformed to the grid frame and determine the spin alignment angles η and δ along each line of sight; η lies in the grid plane and is measured anticlockwise from the projection of the disc symmetry axis; δ is the out-of-plane angle. The local dichroism of the medium is taken as per unit optical depth with components and , which determine the increments of Q and U from emission or absorption along the line of sight. The polarization transfer is taken at first order only, neglecting cross products (e.g. Martin 1974), an approximation which is good up to per cent polarization, and is larger than has been observed. As a check on the above, some of the models were run in which cross-product terms were included (e.g. Lee & Draine 1985, appendix, or equivalently Martin 1974, appendix), and the cross-section C_ph for the phase difference between the orthogonal E vectors is found (e.g. Draine & Lee 1984). The linear polarizations were not discernably affected at the optical depths considered here and circular polarizations were all well below 0.2 per cent. Positive Q is taken as the projection of the disc symmetry axis on the grid plane and positive U at 45° anticlockwise.

The output consists of determinations I, Q, U over these rectangular grids. The grid spacing is varied between the mid- and far-infrared to include the bulk of the emission within the particular waveband. Only a sample of the output is displayed here. At far-infrared and submm wavelengths a grid interval of 40 scale units is used, which encompasses the whole of the model disc. With a ‘standard’ source of 10⁴ L_⊙ at 1 kpc, this corresponds to a grid spacing of about 0.8 arcsec on the sky. For the mid-infrared vector plots a smaller grid interval is used in order to resolve the luminous region; here the grid spacing is 5 scale units or about 0.1 arcsec.

Field configurations

The model has been run with a variety of field configurations:

(I) pure axial, constant;

(II) pure azimuthal (toroidal), constant

(III) pinched axial, or ‘hourglass’, constant,

(IVa,b) ‘Königl’ self-similar type with radial, azimuthal and axial components – approximating a Keplerian disc with accretion, , B_z constant;

(V) Königl with just azimuthal and axial components – e.g. Keplerian without inflow, , B_z constant;

(VI) a helically twisted field, , B_z constant;

(VII) a dipole field.

Here

 

These seven field models are not intended to represent any theoretical field model, but are merely indicative of the kinds of fields which may play a role in disc development. We consider the grains to be uniformly aligned so that only the direction and not the magnitude of B is of interest and the polarization configurations of models I, II and VII are therefore completely defined.

In contrast models (III)–(VI) have a continuous parameter range. γ determines how much an axial field is pinched in the mid-plane, but note that for a 30° flare it is not possible to produce a strong pinch with the simple field description we have used: if the field will not emerge from the face of the disc so we have chosen a value of for the models. α determines how tightly the disc field is wound up: for the field lines twist through one radian between the mid plane and their exit from the 30° disc and the change from an axial field is not large. In models IV and V we have chosen two values of α (40 and 2) so that in the first case field lines twist by more than one turn (1.75) before their exit and in the second by 1/2 radian. The equations of the field lines are

 

and

 

where r₀ and φ₀ are the mid-plane values. For a 30° flare the exit values are , and .

Three of these configurations are shown diagrammatically in Fig. 1 which shows particular field lines for cases III, IVa and IVb and their relation to a partially cut-away disc. The line is shown from its entry to its exit from the disc: solid in the cut-away section and dotted in the ‘interior’.

Figure 1

A cut-away section of the flared disc showing three of the field structures used. For the pinched axial field (III) and the pinched and weakly twisted field (IVa) four field lines are shown, solid in the cut-away sections and dotted inside. For the more strongly twisted field only one field line is shown. The lines are shown only from entry to exit from the face of the disc.

Apart from (II) and (VII) all the other field configurations can be regarded as essentially perturbed poloidal (taken here to mean open field lines rather than closed as in toroidal) and all but II and VI are purely axial on the median plane.

Results

Surface brightness

In this treatment the small dependence of radiative transfer on grain alignment is neglected and to this approximation the temperature distribution for a given density law is common to all the field models I–VII, and at a given wavelength and view angle the surface brightness distribution will also be common. In the mid-infrared the surface brightness is dominated by the warm central region and the brightest few decades of flux are contained within less than 10 model radial units from the centre at 10 μm or about 2 arcsec for our ‘standard’ source. In Fig. 2, and subsequent mid-infrared plots, samples at four wavelengths are shown, at 10, 13, 18 and 22 μm anticlockwise from the top left; in each case the highest contour is a factor of three less bright than the peak and subsequent contours represent reduction by a further factor of three. Bearing in mind the steep brightness gradient, these images approach the limit of resolvability even with 10-m class telescopes. There is only a small gain over the diffraction limit in moving to longer wavelengths than 10 μm but the chief advantage is the gain in flux in avoiding the silicate absorption.

Figure 2

Model I. Polarization images at 10, 13, 18 and 22 μm (anticlockwise from top left of each panel) and spectropolarimetry from of field models I–VI at three view angles. Brightness contours are separated by a factor of 3 and the highest contour is at 1/3 of the peak brightness.

In the submm, sampled here at 850 μm, the fall in brightness from the maxima to the disc edge is roughly an order of magnitude for a uniform disc and limb brightening and a central minimum is very pronounced when viewed face-on. At small view angles to the disc plane there is a bright bar along the major axis, widening towards the edges, and at intermediate angles there is a strong central minimum with two equispaced maxima along the major axis. These features are displayed in Fig. 3 and subsequent submm plots, in which the contours are linearly spaced. With a radial density dependence of form r^−1.5, much of this structure is lost in the steeper fall-off in brightness from the centre, but the centre-to-edge brightness ratio is still not much more than an order of magnitude.

Figure 3

Polarization images at 850 μm of field model II at different view angles, i. Brightness contours are linearly spaced: from to 30° (flare angle) the contours increase towards the centre, beyond this angle a central minimum develops with two peaks on either side normal to the projection of the disc axis. Near face-on views show the brightness increasing from a central minimum to a sharp fall-off near the rim.

Polarization

For the range of A_V up to several hundred magnitudes the polarization is emissive in the far-infrared and submm and in the latter effectively optically thin. The magnetic field direction is then orthogonal to the E vector of polarization, and so long as the emission is optically thin the whole depth of the disc is sampled. In these spectral regions the whole spatial extent of the model disc is observed as well. These properties make polarimetric images at the longer wavelengths a less complex and more straightforward representation of the field distribution than from the mid-infrared but with much poorer spatial resolution, at least until synthesis systems become available. Summing of the entire line of sight is not necessarily helpful, however. For example frequently the U component changes sign between the near and far side of the disc and for edge-on views this information cancels out with polarizations then tending to be parallel to either the disc axis or median plane.

In the mid-infrared the observable region shrinks, and only lines of sight originating from the central warm regions give information on the field distribution. Usually there is a marked change of polarization structure according to whether the view angle is greater or less than the disc flare angle. This is largely an optical depth effect depending on whether the central mid-infrared emitting regions are viewed through the cooler outer absorbing regions of the disc or not. In practice in this wavelength region some spectropolarimetry will be an essential diagnostic tool to separate the roles of absorption and emission; in the former case the silicate absorption features and their sharply peaked polarization profiles are prominent and their position angles indicate the field directions projected on the plane of the sky. If emissive polarization dominates, the silicate absorption feature is absent and the sharp polarization peak at 10 μm is replaced by a broad maximum near 12 μm; the position angles become orthogonal to the projected field as at longer wavelengths (Fig. 2). The occurrence of a strong absorption feature appears to be a necessary but not sufficient condition for the polarization to show a strong peak just longwards of the absorption maximum: for the latter to appear the fields in the cool outer regions must have a large component transverse to the line of sight. The polarization peak from absorption is shifted to a slightly longer wavelength than the absorption maximum. This is a consequence of dichroism and is a diagnostic that the peak is not a result of other effects such as absorption in the silicate feature of an unpolarized diluting source or component in the beam (e.g. Elsasser & Staude 1978).

Note that for a density law r^−1.5 and , emissive polarization dominates at all view angles and .

Intermediate-wavelength images at 100 and 350 μm have also been run: these cover essentially all of the disc with substantially the same polarization structures as at 850 μm and are not presented here.

Some representative polarization maps at mid-infrared wavelengths and at 850 μm with examples of mid to far-infrared spectropolarimetry are shown in Figs 2–12 and 14–17, and a discussion of some general properties follows; note that in all cases E vectors of polarization are shown. The spectral plots sample the whole disc but the mid-infrared profiles arise only from the inner regions, i.e. are what would be observed in small, few arcsec, beams. In many of the spectral plots the position angles are either 0° (coinciding with the wavelength axis) or 90°. A change between these angles is often abrupt and a steep linear rise between them indicates that the transition has occurred between consecutive wavelength samples.

Figure 4

As Fig. 2 for field model II.

Figure 5

As Fig. 2 for field model III.

Figure 6

As Fig. 3 for field model III.

Figure 7

As Fig. 2 for field model IVa.

Figure 8

As Fig. 3 for field model IVa.

Figure 9

As Fig. 3 for field model IVb.

Figure 10

As Fig. 2 for field model IVb.

Figure 11

As Fig. 2 for field model V.

Figure 12

As Fig. 3 for field model V.

Figure 14

As Fig. 2 for field model VI.

Figure 15

As Fig. 3 for field model VI.

Figure 16

As Fig. 3 for field model VII.

Figure 17

As Fig. 2 for field model VII.

I: axial

Here the position angle, θ, of absorptive polarization seen in the mid-infrared mirrors the uniform axial field and shows only a change of magnitude with increasing view angle, i, up to the flare angle; the polarization is large and of order per cent at 10 μm and ∼3 per cent at 18 μm in the silicate features, but much smaller at 13 and 22 μm. Beyond the flare angle the polarization shifts to being primarily emissive and just a few per cent, almost independent of wavelength, and its position angle is shifted through 90° to become orthogonal to the disc symmetry axis and lies along the major axis of the disc image. The polarization spectral and position angle changes across this transition are shown in Fig. 2, but note that the wavelength of the transition depends quite strongly on the value of A_V, here 50 mag, for the central extinction. For values of A_V much less than this there can be a region of 90° position angle in the region of 15 μm as well and the polarization spectrum becomes a confusing mixture of emission and absorption.

In the present case emissive polarization dominates beyond about 25 μm for all view angles, and in this long-wavelength regime the polarization is uniform across the image, with an amplitude that varies as cos²i with position angle parallel to the major axis of the disc image. For a uniform alignment these statements remain true for all density dependences so long as the line of sight remains optically thin. This is the most straightforward polarization configuration.

II: pure azimuthal or toroidal

The polarization distributions from an azimuthal field are also easily understood in terms of emission and absorption in the mid-infrared, resulting in position angles lying closely parallel to the disc image major axis for i up to the flare angle (absorption) and becoming radial for larger values of i (emission). For i small and less than the flare the polarization spectral profile is similar to that arising from an axial field and remains characteristic of absorption to a slightly longer wavelength of about 35 μm than for the axial case; it is independent of view angle up to the flare angle (Fig. 4). Above this angle the emissive polarizations become radial, but unless this small region is resolved the observed polarization will be very small owing to cancellation of the Stokes vectors.

Longwards of 35 μm or so, the polarization is predominantly axial for small i, with amplitude reducing with increasing distance from the centre along the major axis of the disc image (Fig. 3). This is because along these lines of sight the transverse components of field are small. The region of maximum polarization is centred on the disc axis and becomes wider as the distance from the mid plane increases with the polarization amplitude forming an ‘X’ shaped distribution, owing to the increase of transverse field components along these lines of sight. As i increases and becomes more face on the polarization becomes radial and essentially uniform in amplitude.

III: pinched axial or ‘hourglass’, γ= 2

The polarization spectrum and its position angle is very similar to the case for an axial field so long as the view angle, i, is not greater than about 15°. When i is greater than this the polarization fraction decreases drastically, loses the large 10 μm feature, although strong extinction is still present, and develops a circumferential component that persists from through the flare to a face-on view (Fig. 5). The explanation that the polarization spectrum appears emissive while the extinction persists is that for this configuration the fields in the cool outer regions are mainly longitudinal along these lines of sight, as can be seen in Fig. 1.

In the far-infrared and submm the polarization vectors appear convex to the image major axis for small values of i, as expected, and become circumferential at large values. Not so obviously, in the middle range of view angles the polarization is reduced along the major axis and minima appear at the positions of the two surface brightness maxima (Fig. 6). This is the result of pronounced changes of field directions along these lines of sight: note the roughly orthogonal field lines in projection in Fig. 1 for this situation. In the r^−1.5 density models these polarization minima persist, although the brightness maxima are lost in the steep brightness gradient. Nearly face-on views show limb enhancement of the polarization fraction.

IV: a twisted and pinched field

This kind of configuration is one that has been frequently invoked in rotating disc systems (e.g. Wardle & Königl 1993; Pudritz & Norman 1986; Shu et al. 1988). These are essentially axial fields distorted by rotational shear to have a φ dependence and to be centrally pinched through inflow so that three field components are involved. Here we consider both a fairly strongly wound-up configuration in which the azimuthal component is substantially larger than the radial , and one in which these components are comparable .

• (i)

  From the mid-infrared through the far-infrared and submm and for view angles above the flare this less wound-up configuration shows a clear spiral pattern, and should be readily distinguishable from those so far considered (Figs 7, 8). However for edge on views at long wavelengths it becomes increasingly difficult to distinguish both the image and the polarization profiles from a pinched axial type (III). At the mid range of view angles, ∼, there appear symmetric nulls in the polarization (Fig. 8) positioned at an angle to the disc axis and this might prove significant in diagnosing the field parameters and/or the view angle. These nulls are not apparent for .

• In the mid-infrared the extreme edge-on view is indistinguishable from the axial and pinched axial models in both spectrum and polarization image. For view angles between edge-on and the flare the spectrum retains its absorptive characteristics in polarization but the position angle departs clockwise from 180° (for positive α) to about 110° when (Fig. 7). Additionally the position angle shows smooth variation with wavelength of order 20° through the mid-infrared owing to the relative strengths of emission and absorption through the resonance features.

• (ii)

  . First we consider the far-infrared and submm region (Fig. 9). At all view angles the polarization images are very similar to a purely azimuthal field. This is a result of the large value of α, which ensures that the azimuthal field component exceeds the axial and radial except for regions very close to the median plane. Rather weak minima, or nulls, appear normal to the projection of the disc axis at small inclinations. (Note that the purely azimuthal field II has no axial component.)

At mid-infrared wavelengths the polarization is quite similar to the azimuthal case, with the interesting exception of small view angles to the disc plane. For an extreme edge-on view the polarization from the mid-plane is normal to the plane, and above or below the plane it turns through an angle depending on the distance from the plane; the pattern is reflected in the mid plane (Fig. 10). The sense in which the vectors turn (i.e. to right or left) indicates the sign or handedness of α, but this structure is on a small scale of the order of a few per cent of the extent of the disc, or an arcsec for our ‘standard’ source; observations that do not resolve this region would record a polarization normal to the plane. A curious feature of this extreme view is that there is no change of position angle with wavelength from the mid-infrared to the submm even though the polarization shifts from absorption to emission. This odd effect is a result of absorption primarily along a central line of sight in the mid-infrared, which sees a strong axial component, whereas emission at long wavelengths samples a more extended region where there are large azimuthal field components; in both cases the polarization is axial.

As i departs from zero, however, the position angle in the mid-infrared turns clockwise (α positive) rapidly and by 5° inclination to the disc plane the mid-infrared pattern has become almost uniform across the image with a position angle <15° from the major axis of the image. Even with such a small value of i the azimuthal field components dominate along the line of sight to the central regions and such a configuration would be hard to distinguish from a purely azimuthal field. For larger values of i the polarization position angle approaches 90° and is indistinguishable from a purely azimuthal field, retaining the characteristic polarization profile so long as i≤ the flare angle. Beyond this the polarization is emissive and similar to that at longer wavelengths. These effects show that the observations of high-mass, heavily obscured protostars (Aitken et al. 1993; Holland et al. 1996) in which the polarization indicates fields in the plane of the disc, are the natural consequence of wound-up disc fields viewed through the disc plane, as implied in these cases by the large observed extinction. Only extreme edge-on views would display an axial field and such configurations are statistically infrequent. At larger view angles than the flare angle the polarization is always emissive and radial and straightforward to understand in terms of the field distribution.

V: a twisted field without pinch

This might be considered a subset of IV: a rotationally sheared axial field without a radial component. For the strongly twisted case the polarization image and spectrum are insignificantly different from case IVb at all wavelengths, and in the mid-infrared for they are in turn very similar to the weak case IVa (Fig. 11). The position angle changes with wavelength through the mid-infrared, for view angles below the flare, are more pronounced than for case IVb and continue slowly through 100 μm as the emission becomes optically thin. At longer wavelengths too there are similarities in the polarizations from models IVa and V, and the most noticeable differences are in edge-on views where the inferred field now appears purely axial, without convexity, and nearly face-on when the pattern is uniformly radial and indistinguishable from azimuthal. At intermediate angles the polarization is confused by nulls and large changes of polarization in the central regions (Fig. 12).

Fig. 13 shows the relationship between observed position angle and view angle for unresolved mid-infrared observations and a range of the twist parameter, α.

Figure 13

The dependence of 10-μm polarization position angle with respect to the disc axis as a function of view angle for field models with a twist component defined by α where .

VI: a helical field

A helically twisted field has also been included. Although this is probably not expected to play an important role in the star formation process, it may appear as distortions in the larger scale ambient field superimposed upon the region and through the disc plane. In the present case we take the helical axis to be the disc axis and consider a helix, the periodicity of which parallel to its axis is independent of radial distance: a simple twist, in this case of between the top and bottom at the rim of the disc. Such a twist approximates to the maximum effects of helicity, because larger twists will asymptote towards toroidal fields and weaker twists towards axial.

In the mid-infrared, views within the flare angle show uniform polarization at ∼45° to the axis (Fig. 14) and the position angle changes towards 90° in the far-infrared with variations through the mid-infrared resonances. Above the flare the mid-infrared sees only the central thin region of the disc where the twist angle is small and the fields tend to be axial: the polarization image then resembles that for a simple axial field.

In the far-infrared and submm the images from above the flare show radial polarization with the polarization amplitude maximizing on one side of the image. As the view angle increases towards face-on, the polarization amplitude minimizes towards the centre, as for the mid-infrared. For the small view angles within the flare the polarization depends strongly on the pitch of the helix: in the present case the polarization is parallel to the image major axis in the extreme edge-on view and develops two regimes of nearly constant position angles tilted to and reflected in this axis as the view angle is increased (Fig. 15). The polarization maximum is again shifted to one side of the image.

VII: a dipole field

In near face-on views (>75°) at long wavelengths this configuration gives a simply circumferential polarization. Near a 45° view angle the long-wavelength polarization becomes predictably elliptical in profile, matching the surface brightness outline, but nulls appear close to the intensity maxima on the disc axis and these polarization minima seem to extend in arcs towards the poles (Fig. 16). These nulls remain with a steep radial density dependence. At angles within the flare the normals to the long-wavelength polarization trace out a dipole field pattern with the appearance of minima along the disc axis towards the rim.

Little of this structure is evident in the mid-infrared within the flare angle and the polarization is simply axial. Above the flare the mid-infrared polarization is small and crab-like (Fig. 17).

Effect of density dependence

A natural effect of the r^−1.5 density law is the contraction of the emitting region, noticeable at all wavelengths but particularly so in the far-infrared; the polarization is little changed from the uniform density case. In the mid-infrared, however, there are changes to the polarization pattern, absorptive polarization being less prominent especially in the outer regions of the disc, and the overall polarization at small view angles is smaller than in the constant density case. The mid-infrared polarization features are very subdued and probably not identifiable from spectropolarimetry, although 90° position angle changes can occur. Through the far-infrared and at all wavelengths for view angles larger than the flare, the polarization pattern is little changed from the constant density case. It is noticeable that in spite of the steep density dependence, which washes out surface brightness structure, the associated polarization structure remains but at very low surface brightness.

Effect of non-aligned grain components and regions

In most of the field distributions considered here the inclusion of a non-aligned grain component of different size and/or chemistry results in little change to the position angle of the polarization but the polarization fraction is diluted and often redistributed in wavelength. For instance if half the 0.06-μm aligned grains are replaced with non-aligned smaller grains the far-infrared and submm polarization is roughly halved, as might be expected, but the polarization in the mid-infrared silicate features is drastically reduced and can almost vanish.

Confinement of the radial extent of alignment to the inner regions has a much more dramatic effect and restricts the polarization to a small inner region and to or so; polarization at longer wavelengths is essentially diluted away and the mid-infrared polarization is predominantly emissive. On the other hand if only the outer, cooler, regions of the disc contain aligned grains then the polarization is little different from the uniformly aligned case.

Circular polarization

Maximum circular polarization occurs at a wavelength of , and when there is a significant twist of alignment along the line of sight. In all cases considered here it is <0.2 per cent and probably not unambiguously detectable with current instrumentation. It is very dependent on optical depth and could exceed 1 per cent for .

Interpretation of polarimetry

First there are some observational limitations to consider. Our ‘standard’ source of ∼40 arcsec at 850 μm will be poorly sampled with existing single dishes. In the mid-infrared the source has a steep brightness fall-off of a factor ∼30 within ∼ few arcsec; this makes spatially resolved polarimetry difficult, certainly with a single-beam instrument, even on 10-m class telescopes. It appears that the greatest advances in the submm will be when synthesis systems such as ALMA are operational, and until then we will probably have to depend largely on unresolved spectropolarimetry in the mid-infrared.

At first sight it may seem that there is a confusing variety of polarization structures displayed in Figs 2–12 and 14–17 and that there will be more than a little redundancy in associating images with a field structure. It is worth introducing some kind of taxonomy to try to classify the images in terms of identifiable qualities. Assuming that the mid-infrared can provide little more than spectropolarimetry of an unresolved source, its main characteristics will be (i) the flux spectrum and the appearance or absence of extinction, (ii) the polarization spectrum and whether it is typical of emission or absorption, and (iii) the position angle spectrum behaviour through the mid-infrared resonances and how it relates to the far-infrared and submm position angle. In the submm we can divide the images into broad categories related to the presence of symmetries about any axes (if any are apparent or known) or whether radial, circumferential or spiral features are apparent.

Mid-infrared spectropolarimetry

Here θ denotes position angle, ‘pabs’ denotes a sharp polarization peak just beyond 10 μm (and at 18 μm), ‘pem’ a weakly structured polarization spectrum typical of emission.

Silicate extinction, pabs, θ constant

With θ parallel to disc symmetry axis, there are several possibilities:

• (I i < flare);

• (III i ≲ 20°

• (IVa if nearly edge on);

• (V i ≲ 5°

• (VI i < flare);.

• With θ normal to disc axis, there are two possibilities:

• (II i < flare);

• (IVb i < flare, excluding edge-on).

• For all of the above there is an abrupt shift of π/2 in position angle to the FIR. If θ is not aligned with axes, see next section.

Silicate extinction, pabs, θ varying through the MIR

If θ is not obviously associated with either axis, there are three possibilities:

• (IVa , excluding edge-on);

• (V

• (VI .

Silicate extinction, pem, θ strongly varies through MIR

• (III

No extinction feature, pem or no polarization

This suggests that the disc is viewed from above the flare angle.

θ parallel to the disc axis:

(II) or (IVb)

θ normal to the disc axis:

(I) or (III) or (IVa) or (V) or (VI) or (VII)

Resolved polarimetric imaging could reduce this redundancy.

Far-infrared and submm imaging polarimetry

Given enough resolution, this can be much more discriminating. The polarimetric structures can be categorized according to whether they show radial or circumferential structure or to the symmetry properties of the polarization (fraction and position angle) on reflection about one or two axes usually defined by the image brightness structure. Many of the images show mirror reflection of polarization, but in some cases the vectors retain their orientation as if transposed across the axis.

In what follows M denotes the image major axis, N its minor axis, p the polarization fraction, and θ its position angle with respect to the image.

Mirror symmetric

 

Approximate rotational symmetry

 

Others

 

Implications for polarimetry

From Section 5.2 it is clear that often a knowledge of the projection angle of the disc axis on the sky is needed; in the FIR and submm this may come from the brightness profiles but mostly will not be evident from the MIR observations alone. Constraints on the viewing angle, density distribution and grain properties can also be obtained from radiative transfer modelling of the spectral energy dependences (SEDs) of these objects. However, even without this information there appear certain polarization characteristics where a particular field model is favoured. As an example, a circumferential field with limb-enhanced polarization would indicate a pinched axial field viewed at a large angle to the disc plane; weak emissive polarization in the MIR would support this view. Also, if MIR spectropolarimetry indicates strong extinction and prominent polarization absorption features together with a constant position angle normal to the (known) projected disc axis, then either a strongly twisted or an azimuthal field is indicated; usually the twisted option is assumed as this can be derived from an external poloidal field and can arise from a wide range of α in the models. This is shown in Fig. 13 and is the probable reason for the frequent indication of fields lying in the plane of the apparent disc (Aitken et al. 1993; Holland et al. 1996). Unfortunately these two configurations will be indistinguishable from FIR and submm images. Promisingly, so long as some of observed polarization morphologies are identifiable in terms of Section 5.3, at least some constraints can be placed on the possible field configurations.

In the nature of things the structures and fields considered here are idealized approximations to those likely to be involved in star formation. Indeed the putative discs themselves are usually observed merely as elongated structures associated with outflows, occasionally with evidence for rotation, and there may be little information on view angle to the disc plane. In addition the disc will have evolved from and sometimes still be embedded within its parent molecular cloud, which also has a magnetic field structure. Aside from these difficulties, which will only be settled by further observations, an encouraging feature in running these models has been that the polarization structure in the submm is nearly independent of density structure. Emissive polarization at these wavelengths is independent of optical depth and so long as grain alignment does not depend strongly on density or field strength the results should be independent of the radial density behaviour or clumpiness of the dust.

Early results at FIR and submm wavelengths have not so far yielded a great deal in the way of identifiable polarization structures; this should not be a surprise in view of the currently attainable resolutions, which are barely able to resolve the regions of interest. Nevertheless some recent results do show some structures suggestive of some of the models presented here: Momose et al. (2001) draw attention to two low polarization regions on either side of NGC 7538 IRS1 and the observations of Chrysostomou et al. (2001) reveal similar symmetric polarization nulls about two components of W51. Such symmetric polarization minima occur in the present field models when a predominantly axial field is twisted within a disc as in IVa (Fig. 8, , IVb (Fig. 9, , and V (Fig. 12, , and the appearance of these symmetries about the source seems to be associated with an additional toroidal component in an axial field distribution. Note, however, that the line on which the minima lie is not directly related to the disc axis unless the degree of twist is strong, as in IVb, where the minima define a line normal to the disc symmetry axis.

A further noticeable trend is that the polarization amplitude at FIR and submm wavelengths tends to be smaller in the central bright and dense regions (e.g. Gonatas et al. 1990; Dotson et al. 2000; Chrysostomou et al. 2001). In a few sources, such as OMC-1, a part or all of this reduction is clearly associated with unresolved source complexity (Aitken et al. 1997) and many of the models presented here similarly exhibit low polarization in the central regions when convolved with a larger beam than shown in the figures. For example if the strongly twisted field model, IVb, at an inclination angle 60° (Fig. 9), is integrated within a circular radius of 280 scale units (corresponding to a beam diameter of 11 arcsec for our ‘standard’ source) the central polarization is rather less than 1 per cent while that found at a radius of 360 units from the centre varies between 2.8 and 3.6 per cent. Similar results are found for other models (II, III, IV, V and VII), which show strong position angle structure in their central regions when viewed at large inclinations.

The observed frequency of low polarization hotspots at long wavelengths is very much what is to be expected at current ∼ resolutions, and gives promise that the increase of resolution expected with synthesis systems such as ALMA will lead to an exciting phase in the study of star formation.

Conclusions

Polarization images from a variety of magnetic field configurations within flared discs which are internally heated are presented and discussed in terms of their ability to discriminate between the field models. It is found that a number of these models should be clearly identifiable and constraints placed on the presence of others. There are likely to be problems from source multiplicity, foreground extinction and polarization, and some uncertainties regarding alignment strength, but the seriousness of these will only become apparent through observations of real sources. Even if the disc fields are chaotic, as Balbus & Hawley (1998) suggest it will be useful to establish their trend with respect to the disc axis.

Imaging polarimetry through the infrared to submm and mid-infrared spectropolarimetry promises to give new insights into magnetic field structures, in particular in star-forming regions, especially when its full potential is realized with submm synthesis systems.

Acknowledgments

We wish to thank Antonio Chrysostomou for helpful discussions, and an unknown referee for constructive suggestions.